properties of hyperreals under field operations
Let ${}^{*}\mathbb{R}_{b}$ denote the set of finite (or limited) hyperreal numbers and ${}^{*}\mathbb{R}_{0}$ the set of infinitesimal^{} hyperreal numbers.
 We have that

1.
${}^{*}\mathbb{R}_{b}$ and ${}^{*}\mathbb{R}_{0}$ are subrings of ${}^{*}\mathbb{R}$.

2.
${}^{*}\mathbb{R}_{0}$ is an ideal of ${}^{*}\mathbb{R}_{b}$.

3.
the sum of an infinite^{} hyperreal with a finite hyperreal is infinite.

4.
the inverse^{} of a nonzero infinitesimal hyperreal is infinite.

5.
the inverse of an infinite hyperreal is infinitesimal.
The above properties can be described more informally like:

1.
finite $+$ finite $=$ finite

2.
infinitesimal $+$ infinitesimal $=$ infinitesimal

3.
infinite $+$ finite $=$ infinite

4.
finite $\times $ finite $=$ finite

5.
infinitesimal $\times $ finite $=$ infinitesimal

6.
infinitesimal${}^{1}$ $=$ infinite

7.
infinite${}^{1}$ $=$ infinitesimal
Title  properties of hyperreals under field operations 

Canonical name  PropertiesOfHyperrealsUnderFieldOperations 
Date of creation  20130322 17:26:19 
Last modified on  20130322 17:26:19 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  4 
Author  asteroid (17536) 
Entry type  Result 
Classification  msc 26E35 